Fibrations and log-symplectic structures

TL;DR

This paper explores the existence and construction of log-symplectic structures on manifolds using Lie algebroid techniques, introducing new concepts like $b$-hyperfibrations and connecting them to other geometric structures.

Contribution

It introduces the notion of $b$-hyperfibrations and demonstrates how they generate log-symplectic structures, linking them to achiral Lefschetz fibrations and folded-symplectic forms.

Findings

Log-symplectic structures can be constructed via $b$-hyperfibrations.

Connections established between log-symplectic, achiral Lefschetz, and folded-symplectic structures.

Abstract

Log-symplectic structures are Poisson structures $\pi$ on $X^{2n}$ for which $\bigwedge^n \pi$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps. More precisely, we introduce the notion of a $b$-hyperfibration and show that they give rise to log-symplectic structures. Moreover, we link log-symplectic structures to achiral Lefschetz fibrations and folded-symplectic structures.